Calculation of Rozansky-Witten Invariants

on the Hilbert Schemes of Points on a K3 Surface

and the Generalised Kummer Varieties

Marc A. Nieper-Wißkirchen^{1}

Received: September 1, 2003 Revised: December 19, 2003 Communicated by Thomas Peternell

Abstract. For any holomorphic symplectic manifold (X, σ), a closed Jacobi diagram with 2k trivalent vertices gives rise to a Rozansky-Witten class

RW_{X,σ}(Γ) ∈ H^{2k}(X, O_{X}).

If X is irreducible, this defines a number β_{Γ}(X, σ) by RW_{X,σ}(Γ) = β_{Γ}(X, σ)[σ̄]^{k}.

Let (X^{[n]}, σ^{[n]}) be the Hilbert scheme of n points on a K3 surface together with a symplectic form σ^{[n]} such that ∫_{X^{[n]}}(σ^{[n]}σ̄^{[n]})^{n} = n!. Further, let (A^{[[n]]}, σ^{[[n]]}) be the generalised Kummer variety of dimension 2n − 2 together with a symplectic form σ^{[[n]]} such that ∫_{A^{^{[}^{[n]}^{]}}}(σ^{[[n]]}σ̄^{[[n]]})^{n} = n!. J. Sawon conjectured in his doctoral thesis that for every connected Jacobi diagram, the functions β_{Γ}(X^{[n]}, σ^{[n]}) and β_{Γ}(A^{[[n]]}, σ^{[[n]]}) are linear in n.

We prove that this conjecture is true for Γ being a connected Jacobi diagram homologous to a polynomial of closed polywheels. We further show how this enables one to calculate all Rozansky-Witten invariants of X^{[n]} and A^{[[n]]} for closed Jacobi diagrams that are homologous to a polynomial of closed polywheels. It seems to be unknown whether every Jacobi diagram is homologous to a polynomial of closed polywheels. If indeed the closed polywheels generate the whole graph homology space as an algebrea, our methods will thus enable us to compute all Rozansky-Witten invariants for the Hilbert schemes and the generalised Kummer varieties using these methods.

Also discussed in this article are the definitions of the various graph homology spaces, certain operators acting on these spaces and their relations, some general facts about holomorphic symplectic manifolds and facts about the special geometry of the Hilbert schemes of points on surfaces.

1The author is supported by the Deutsche Forschungsgemeinschaft. This work has been written under hospitality of the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge.

2000 Mathematics Subject Classification: 53C26, 14Q15, 57M15, 05C99

Contents

1. Introduction 592 2. Preliminaries 594 2.1. Some multilinear algebra 594 2.2. Partitions 595 2.3. A lemma from umbral calculus 596 3. Graph homology 597 3.1. The graph homology space 597 3.2. Operations with graphs 600 3.3. An 𝔰𝔩_{2}-action on the space of graph homology 603 3.4. Closed and connected graphs, the closure of a graph 604 3.5. Polywheels 606 4. Holomorphic symplectic manifolds 606 4.1. Definition and general properties 606 4.2. A pairing on the cohomology of a holomorphic symplectic

manifold 607

4.3. Example series 608 4.4. About α^{[n]} and α^{[[n]]} 609 4.5. Complex genera of Hilbert schemes of points on surfaces 611 5. Rozansky-Witten classes and invariants 614 5.1. Definition 614 5.2. Examples and properties of Rozansky-Witten classes 615 5.3. Rozansky-Witten classes of closed graphs 616 6. Calculation for the example series 617 6.1. Proof of the main theorem 617 6.2. Some explicit calculations 620 References 622

1. Introduction

A compact hyperkähler manifold (X, g) is a compact Riemannian manifold whose holonomy is contained in Sp(n). An example of such a manifold is the K3 surface together with a Ricci-flat Kähler metric (which exists by S. Yau’s theorem [18]). In [15], L. Rozansky and E. Witten described how one can associate to every vertex-oriented trivalent graph Γ an invariant b_{Γ}(X) to X, henceforth called a Rozansky-Witten invariant of X associated to Γ. In fact, this invariant only depends on the homology class of the graph, so the invariants are already defined on the level of the graph homology space B (see e.g. [1] and this paper for more information about graph homology).

Every hyperkähler manifold (X, g) can be given the structure of a Kähler manifold X (which is, however, not uniquely defined) whose Kähler metric is just given by g. X happens to carry a holomorphic symplectic two-form σ ∈ H^{0}(X, Ω_{X}^{2}), whereas we shall call X a holomorphic symplectic manifold. Now M. Kapranov showed in [8] that one can in fact calculate b_{Γ}(X) from (X, σ) by purely holomorphic methods. The basic idea is the following: We can identify the holomorphic tangent bundle T_{X} of X with its cotangent bundle Ω_{X} by means of σ. Doing this, the Atiyah class α_{X} (see [8]) of X lies in H^{1}(X, S_{3}T_{X}). Now we place a copy of α_{X} at each trivalent vertex of the graph, take the ∪-product of all these copies (which gives us an element in H^{2k}(X, (S_{3}T_{X})^{⊗2k}) if 2k is the number of trivalent vertices), and finally contract (S_{3}T_{X})^{⊗2k}) along the edges of the graph by means of the holomorphic symplectic form σ. Let us call the resulting element RW_{X,σ}(Γ) ∈ H^{2k}(X, O_{X}). In case 2k is the complex dimension of X, we can integrate this element over X after we have multiplied it with [σ]^{2k}. This gives us more or less b_{Γ}(X). The orientation at the vertices of the graph is needed in the process to get a number which is not only defined up to sign. There are two main example series of holomorphic symplectic manifolds, the Hilbert schemes X^{[n]} of points on a K3 surface X and the generalised Kummer varieties A^{[[n]]} (see [2]). Besides two further manifolds constructed by K. O’Grady in [13] and [12], these are the only known examples of irreducible holomorphic symplectic manifolds up to deformation. Not much work was done on actual calculations of these invariants on the example series. The first extensive calculations were carried out by J. Sawon in his doctoral thesis [16]. All Chern numbers are in fact Rozansky-Witten invariants associated to certain Jacobi diagrams, called closed polywheels. Let W be the subspace spanned by these polywheels in B. All Rozansky-Witten invariants associated to graphs lying in W can thus be calculated from the knowledge of the Chern numbers (which are computable in the case of X^{[n]} ([3]) or A^{[[n]]} ([11]). However, from complex dimension four on, there are graph homology classes that do not lie in W. J. Sawon showed that for some of these graphs the Rozansky-Witten invariants can still be calculated from knowledge of the Chern numbers, which enables one to calculate all Rozansky-Witten invariants up to dimension five. His calculations would work for all irreducible holomorphic manifolds whose Chern numbers are known. In this article, we will make use of the special geometry of X^{[n]} and A^{[[n]]}. Doing this, we are able to give a method which enables us to calculate all Rozansky-Witten invariants for graphs homology classes that lie in the algebra C generated by closed polywheels in B. The closed polywheels form the subspace W of the algebra B of graph homology. This is really a proper subspace. However, C, the algebra generated by this subspace, is much larger, and, as far as the author knows, it is unknown whether C = B, i.e. whether this work enables us to calculate all Rozanky-Witten invariants for the main example series. The idea to carry out this computations is the following: Let (Y, τ) be any irreducible holomorphic symplectic manifold. Then H^{2k}(Y, O_{Y}) is spanned by [τ̄]^{k}. Therefore, every graph Γ with 2k trivalent vertices defines a number β_{Γ}(Y, τ) by RW_{Y,τ} (Γ) = β_{Γ}(Y, τ)[τ̄]^{k}. J. Sawon has already discussed how knowledge of these numbers for connected graphs is enough to deduce the values of all Rozansky-Witten invariants. For the example series, let us fix holomorphic symplectic forms σ^{[n]}, respective σ^{[[n]]} with ∫_{X^{[n]}}(σ^{[n]}σ̄^{[n]})^{n} = n! respective ∫_{A^{^{[}^{[n]}^{]}}}(σ^{[[n]]}σ̄^{[[n]]})^{n} = n!. J. Sawon conjectured the following:

The functions β_{Γ}(X^{[n]}, σ^{[n]}) and β_{Γ}(X^{[[n]]}, σ^{[[n]]}) are linear in n for Γ being a connected graph.

The main result of this work is the proof of this conjecture for the class of connected graphs lying in C (see Theorem 3). We further show how one can calculate these linear functions from the knowledge of the Chern numbers and thus how to calculate all Rozansky-Witten invariants for graphs in C. We should note that we don’t make any use of the IHX relation in our derivations, and so we could equally have worked on the level of Jacobi diagrams. Let us finally give a short description of each section. In section 2 we collect some definitions and results which will be used later on. The next section is concerned with defining the algebra of graph homology and certain operations on this space. We define connected polywheels and show how they are related with the usual closed polywheels in graph homology. We further exhibit a natural 𝔰𝔩_{2}-action on an extended graph homology space. In section 4, we first look at general holomorphic symplectic manifolds. Then we study the two example series more deeply. Section 5 defines Rozansky-Witten invariants while the last section is dedicated to the proof of our main theorem and explicit calculations.

2. Preliminaries

2.1. Some multilinear algebra. Let T be a tensor category (commutative and with unit). For any object V in T , we denote by S^{k}V the coinvariants of V^{⊗k} with respect to the natural action of the symmetric group and by Λ^{k}V the coinvariants with respect to the alternating action. Further, let us denote by S_{k}V and Λ_{k}V the invariants of both actions.

Proposition 1. Let I be a cyclicly ordered set of three elements. Let V be an object in T . Then there exists a unique map Λ_{3}V → V^{⊗I} such that for every bijection ϕ : {1, 2, 3} → I respecting the canonical cyclic ordering of {1, 2, 3} and the given cyclic ordering of I the following diagram

\begin{matrix}Λ_{3}V & & Λ_{3}V \\ ↓ & & ↓ \\ V^{⊗3} & ⟶_{ϕ_{∗}} & V^{⊗I} \end{matrix} (1)

commutes, where the map ϕ_{∗} is the canonical one induced by ϕ.

Proof. Let ϕ, ϕ′ : {1, 2, 3} → I be two bijections respecting the cyclic ordering. Then there exist an even permutation α ∈ 𝔄_{3} such that the lower square of the following diagram commutes:

Λ_{3}V  Λ_{3}V
↓  ↓
V^{⊗3} ⟶_{α_{∗}} V^{⊗3}
ϕ↓  ↓^{ϕ′}
V^{⊗I}  V^{⊗I}.

We have to show that the outer rectangle commutes. For this it suffices to show that the upper square commutes. In fact, since α is an even permutation, every element of Λ_{3}V is by definition invariant under α_{∗}. □

2.2. Partitions. A partition λ of a non-negative integer n ∈ ℕ_{0} is a sequence λ_{1}, λ_{2}, … of non-negative integers such that

∥λ∥ := ∑_{i=1}^{∞} iλ_{i} = n. (2)

Therefore almost all λ_{i} have to vanish. In the literature, λ is often notated by 1^{λ_{1}}2^{λ_{2}} . . .. The set of all partitions of n is denoted by P(n). The union of all P(n) is denoted by P := ⋃_{n=0}^{∞} P(n). For every partition λ ∈ P, we set

|λ| := ∑_{i=1}^{∞} λ_{i} (3)

and

λ! := ∏_{i=1}^{∞} λ_{i}!. (4)

Let a_{1}, a_{2}, … be any sequence of elements of a commutative unitary ring. We set

a_{λ} := ∏_{i=1}^{∞} a_{i}^{λ_{i}} (5)

for any partition λ ∈ P. With these definitions, we can formulate the following proposition in a nice way:

Proposition 2. In ℚ[[a_{1}, a_{2}, …]] we have

exp (∑_{i=1}^{∞} a_{i}) = ∑_{λ∈P} \frac{a_{λ}}{λ!} . (6)

Proof. We calculate

exp (∑_{i=1}^{∞} a_{i}) = ∑_{n=0}^{∞} \frac{1}{n!} (∑_{i=1}^{∞} a_{i})^{n} = ∑_{n=0}^{∞} \frac{1}{n!} ∑_{λ∈P,|λ|=n} n! ∏_{i=1}^{∞} \frac{a_{i}^{λ_{i}}}{λ_{i}!} = ∑_{λ∈P} \frac{a_{λ}}{λ!} . (7)

□

If we set

\frac{∂}{∂a_{λ}} := ∏_{i=1}^{∞} \frac{∂^{λ_{i}}}{∂a_{i}^{λ_{i}}} |_{a_{i}=0} , (8)

we have due to Proposition 2:

Proposition 3. In ℚ[[s_{1}, s_{2}, …]][a_{1}, a_{2}, …] we have

\frac{∂}{∂a_{λ}} exp (∑_{i=1}^{∞} a_{i}s_{i}) = s_{λ}. (9)

2.3. A lemma from umbral calculus.

Lemma 1. Let R be any ℚ-algebra (commutative and with unit) and A(t) ∈ R[[t]] and B(t) ∈ tR[[t]] be two power series. Let the polynomial sequences (p_{n}(x)) and (s_{n}(x)) be defined by

(10)  ∑_{k=0}^{∞} p_{k}(x)\frac{t^{k}}{k!} = exp(xB(t))

and

(11)  ∑_{k=0}^{∞} s_{k}(x)\frac{t^{k}}{k!} = A(t) exp(xB(t)).

Let W_{B}(t) ∈ tR[[t]] be defined by W_{B}(t exp(B(t))) = t. Then we have

(12)  ∑_{k=0}^{∞} \frac{xp_{k}(x − k)}{(x − k)} \frac{t^{k}}{k!} = exp(xB(W_{B}(t)))

and

(13)  ∑_{k=0}^{∞} \frac{s_{k}(x − k)}{k!}t^{k} = \frac{A(W_{B}(t))}{1 + W_{B}(t)B′(W_{B}(t))} exp(xB(W_{B}(t))).

Proof. It suffices to prove the result for the field R = ℚ(a_{0}, a_{1}, …, b_{1}, b_{2}, …) and A(t) = ∑_{k=0}^{∞} a_{k}t^{k} and B(t) = ∑_{k=1}^{∞} b_{k}t^{k}. So let us assume this special case for the rest of the proof. Let us denote by f(t) the compositional inverse of B(t), i.e. f(B(t)) = t. We set g(t) := A^{−1}(f(t)). For the following we will make use of the terminology and the statements in [14]. Using this terminology, (10) states that (p_{n}(x)) is the associated sequence to f(t) and (11) states that (s_{n}(x)) is the Sheffer sequence to the pair (g(t), f(t)) (see Theorem 2.3.4 in [14]).

Theorem 3.8.3 in [14] tells us that (s_{n}(x − n)) is the Sheffer sequence to the pair (g̃(t), f̃(t)) with

g̃(t) = g(t)(1 + f(t)/f′(t))

and

f̃(t) = f(t) exp(t).

The compositional inverse of f̃(t) is given by B̃(t) := B(W_{B}(t)):

B(W_{B}(f̃(t))) = B(W_{B}(f(t) exp(t))) = B(W_{B}(f(t) exp(B(f(t))))) = B(f(t)) = t.

Further, we have

Ã(t) := g̃^{−1}(B̃(t))

= (g(B(t))(1 + f(B(t))/f′(B(t))))^{−1} ∘ W_{B}(t) = \frac{A(t)}{1 + tB′(t)} ∘ W_{B}(t),

which proves (13) again due to Theorem 2.3.4 in [14]. It remains to prove (12), i.e. that (\frac{xp_{n}(x)}{x−n}) is the associated sequence to f̃(t). We already know that (p_{n}(x − n)) is the Sheffer sequence to the pair (1 + f(t)/f′(t), f̃(t)). By Theorem 2.3.6 of [14] it follows that the associated sequence to f̃(t) is given by (1+f(d/dx)/f′(d/dx))p_{n}(x−n). By Theorem 2.3.7 and Corollary 3.6.6 in [14], we have

(1 + \frac{f(d/dx)}{f′(d/dx)}) p_{n}(x − n) = p_{n}(x − n) + \frac{1}{f′(d/dx)}np_{n−1}(x − n)

= p_{n}(x − n) + \frac{np_{n}(x − n)}{x − n} = \frac{xp_{n}(x − n)}{x − n},

which proves the rest of the lemma. □

3. Graph homology

This section is concerned with the space of graph homology classes of unitrivalent graphs. A very detailed discussion of this space and other graph homology spaces can be found in [1]. Further aspects of graph homology can be found in [17], and, with respect to Rozansky-Witten invariant, in [7].

3.1. The graph homology space. In this article, graph means a collection of vertices connected by edges, i.e. every edge connects two vertices. We want to call a half-edge (i.e. an edge together with an adjacent vertex) of a graph a flag. So, every edge consists of exactly two flags. Every flag belongs to exactly one vertex of the graph. On the other hand, a vertex is given by the set of its flags. It is called univalent if there is only one flag belonging to it, and it is called trivalent if there are exactly three flags belonging to it. We shall identify edges and vertices with the set of their flags. We shall also call univalent vertices legs. A graph is called vertex-oriented if, for every vertex, a cyclic ordering of its flags is fixed.

Figure 1. This Jacobi diagram has four trivalent vertices v_{1}, …, v_{4}, and two univalent vertices u_{1} and u_{2}, and e is one of its 7 edges.

Definition 1. A Jacobi diagram is a vertex-oriented graph with only uni- and trivalent vertices. A connected Jacobi diagram is a Jacobi diagram which is connected as a graph. A trivalent Jacobi diagram is a Jacobi diagram with no univalent vertices. We define the degree of a Jacobi diagram to be the number of its vertices. It is always an even number. We identify two graphs if they are isomorphic as vertex-oriented graphs in the obvious sense.

Example 1. The empty graph is a Jacobi diagram, denoted by 1. The unique Jacobi diagram consisting of two univalent vertices (which are connected by an edge) is denoted by ℓ.

Figure 2. The Jacobi diagram ℓ with its two univalent vertices u_{1} and u_{2}.

Remark 1. There are different names in the literature for what we call a “Jacobi diagram”, e.g. unitrivalent graphs, chord diagrams, Chinese characters, Feynman diagrams. The name chosen here is also used by D. Thurston in [17]. The name comes from the fact that the IHX relation in graph homology defined later is essentially the well-known Jacobi identity for Lie algebras. With our definition of the degree of a Jacobi diagram, the algebra of graph homology defined later will be commutative in the graded sense. Further, the map RW that will associate to each Jacobi diagram a Rozansky-Witten class will respect this grading. But note that often the degree is defined to be half of the number of vertices, which still is an integer.

We can always draw a Jacobi diagram in a planar drawing so that it looks like a planar graph with vertices of valence 1, 3 or 4. Each 4-valent vertex has to be interpreted as a crossing of two non-connected edges of the drawn graph and not as one of its vertices. Further, we want the counter-clockwise ordering of the flags at each trivalent vertex in the drawing to be the same as the given cyclic ordering.

Figure 3. These two graphs depict the same one.

In drawn Jacobi diagrams, we also use a notation like ⋯ −^{n} ⋯ for a part of a graph which looks like a long line with n univalent vertices (“legs”) attached to it, for example … ⊥⊥⊥ … for n = 3. The position of n indicates the placement of the legs relative to the “long line”.

Definition 2. Let T be any tensor category (commutative and with unit). Every Jacobi diagram Γ with k trivalent and l univalent vertices induces a natural transformation Ψ^{Γ} between the functors

(14)  T → T ,V ↦ S_{k}Λ_{3}V ⊗ S_{l}V

and

(15)  T → T ,V ↦ S^{e}S^{2}V,

where e := \frac{3k+l}{2} which is given by

(16) Ψ^{Γ} : S_{k}Λ_{3}V ⊗ S_{l}V →^{(1)} ⨂_{t∈T} Λ_{3}V ⊗ ⨂_{f∈U} V →^{(2)} ⨂_{t∈T} ⨂_{f∈t} V ⊗ ⨂_{f∈U} V

→^{(3)} ⨂_{f∈F} V →^{(4)} ⨂_{e∈E} ⨂_{f∈e} V →^{(5)} S^{e}S^{2}V,

where T is the set of the trivalent vertices, U the set of the univalent vertices, F the set of flags, and E the set of edges of Γ. Further,

(1) is given by the natural inclusions of the invariants in the tensor products,

(2) is given by the canonical maps (see Proposition 1 and recall that the sets t are cyclicly ordered),

(3) is given by the associativity of the tensor product,

(4) is given again by the associativity of the tensor product, and finally (5) is given by the canonical projections onto the coinvariants.

Definition 3. We define B to be the ℚ-vector space spanned by all Jacobi diagrams modulo the IHX relation

� = � − � (17) and the anti-symmetry (AS) relation

� + � = 0, (18)

which can be applied anywhere within a diagram. (For this definition see also [1] and [17].) Two Jacobi diagrams are said to be homologous if they are in the same class modulo the IHX and AS relation. Furthermore, let B′ be the subspace of B spanned by all Jacobi diagrams not containing ℓ as a component, and let ^{t}B be the subspace of B′ spanned by all trivalent Jacobi diagrams. All these are graded and double-graded. The grading is induced by the degree of Jacobi diagrams, the double-grading by the number of univalent and trivalent vertices. The completion of B (resp. B′, resp. ^{t}B) with respect to the grading will be denoted by B̂ (resp. B̂′, resp. ^{t}B̂). We define B_{k,l} to be the subspace of B̂ generated by graphs with k trivalent and l univalent vertices. B′_{k,l} and ^{t}B_{k} := ^{t}B_{k,0} are defined similarly.

All these spaces are called graph homology spaces and their elements are called graph homology classes or graphs for short.

Remark 2. The subspaces B_{k} of B̂ spanned by the Jacobi diagrams of degree k are always of finite dimension. The subspace B_{0} is one-dimensional and spanned by the graph homology class 1 of the empty diagram 1.

Remark 3. We have B̂ = ∏_{k,l≥0} B_{k,l}. In view of the following Definition 4, B̂′ and B̂ are naturally ℚ-algebras. As ℚ-algebras, we have B̂ = B̂′[[ℓ]]. Due to the AS relation, the spaces B′_{k,l} are zero for l > k. Therefore, B̂′ = ∏_{k=0}^{∞} ⨁_{l=0}^{k} B′_{k,l}.

Example 2. If γ is a graph which has a part looking like ⋯−^{n}⋯, it will become (−1)^{n}γ if we substitute the part ⋯−^{n}⋯ by ⋯−_{n}⋯ due to the anti-symmetry

relation.

3.2. Operations with graphs.

Definition 4. Disjoint union of Jacobi diagrams induces a bilinear map

B̂ × B̂ → B̂, (γ, γ′) ↦ γ ∪ γ′. (19)

By mapping 1 ∈ ℚ to 1 ∈ B̂, the space B̂ becomes a graded ℚ-algebra, which has no components in odd degrees. Often, we omit the product sign “∪”. B, B′, ^{t}B, and so on are subalgebras.

Definition 5. Let k ∈ ℕ. We call the graph homology class of the Jacobi

diagram ◯^{2k} the 2k-wheel w_{2k}, i.e. w_{2} = �, w_{4} = �, and so on. It has 2k univalent and 2k trivalent vertices. The expression w_{0} will be given a meaning later, see section 3.3.

Remark 4. The wheels w_{k} with k odd vanish in B̂ due to the AS relation.

Let Γ be a Jacobi diagram and u, u′ be two different univalent vertices of Γ. These two should not be the two vertices of a component ℓ of Γ. Let v (resp. v′) be the vertex u (resp. u′) is attached to. The process of gluing the vertices u and u′ means to remove u and u′ together with the edges connecting them to v resp. v′ and to add a new edge between v and v′. Thus, we arrive at a new graph Γ/(u, u′), whose number of trivalent vertices is the number of trivalent vertices of Γ and whose number of univalent vertices is the number of univalent vertices of Γ minus two. To make it a Jacobi diagram we define the cyclic orientation of the flags at v (resp. v′) to be the cyclic orientation of the flags at v (resp. v′) in Γ with the flag belonging to the edge connecting v (resp. v′) with u (resp. u′) replaced by the flag belonging to the added edge. For example,

Figure 4. Gluing the two univalent vertices u and u′ of the left graph produces the right one, denoted by �_{2}.

gluing the two univalent vertices of w_{2} leads to the graph �. If π = {{u_{1}, u′_{1}} , …, {u_{k}, u′_{k}}} is a set of two-element sets of legs that are pairwise disjoint and such that each pair u_{k}, u′_{k} fulfills the assumptions of the previous construction, we set

Γ/π := Γ/(u_{1}, u′_{1})/ … /(u_{k}, u′_{k}). (20)

Of course, the process of gluing two univalent vertices given above does not work if u and u′ are the two univalent vertices of ℓ, thus our assumption on Γ.

Definition 6. Let Γ, Γ′ be two Jacobi diagrams, at least one of them without ℓ as a component and U = {u_{1}, …, u_{n}} resp. U′ the sets of their univalent vertices. We define

(21)  Γ̂(Γ′) := ∑_{f:U↪U′ ; injective}(Γ ∪ Γ′)/(u_{1}, f(u_{1}))/ … /(u_{n}, f(u_{n})),

viewed as an element in B̂. This induces for every γ ∈ B̂ a ^{t}B̂-linear map

γ̂ : B̂′ → B̂′, γ′ ↦ γ̂(γ′). (22)

Example 3. Set ∂ := \frac{1}{2} ℓ̂. It is is an endomorphism of B̂′ of degree −2. For example, ∂� = �. By setting

∂(γ, γ′) := ∂(γ ∪ γ′) − ∂(γ) ∪ γ′ − γ ∪ ∂(γ′) (23) for γ, γ′ ∈ B̂′, we have the following formula for all γ ∈ B̂′:

∂(γ^{n}) = \binom{n}{1}∂(γ)γ^{n−1} + \binom{n}{2}∂(γ, γ)γ^{n−2}. (24)

This shows that ∂ is a differential operator of order two acting on B̂′. Acting by ∂ on a Jacobi diagram means to glue two of its univalent vertices in all possible ways, acting by ∂(·, ·) on two Jacobi diagrams means to connect them by gluing a univalent vertex of the first with a univalent vertex of the second in all possible ways.

Definition 7. Let Γ, Γ′ be two Jacobi diagrams, at least one of them without ℓ as a component, and U = {u_{1}, …, u_{n}} resp. U′ the sets of their univalent vertices. We define

(25) ⟨Γ, Γ′⟩ := ∑_{f:U→U′ ; bijective}(Γ ∪ Γ′)/(u_{1}, f(u_{1}))/ … /(u_{n}, f(u_{n})),

viewed as an element in ^{t}B̂. This induces a ^{t}B̂-bilinear map

⟨·, ·⟩ : B̂′ × B̂ → ^{t}B̂, (26)

which is symmetric on B̂′ × B̂′.

Note that ⟨Γ, Γ′⟩ is zero unless Γ and Γ′ have equal numbers of univalent vertices. In this case, the expression is the sum over all possibilities to glue the univalent vertices of Γ with univalent vertices of Γ′. Note that ⟨Γ, Γ′⟩ is zero unless Γ and Γ′ have equal numbers of univalent vertices. In this case, the expression is the sum over all possibilities to glue the univalent vertices of Γ with univalent vertices of Γ′.

Proposition 4. The map ⟨1, ·⟩ : B̂ → ^{t}B̂ is the canonical projection map, i.e. it removes all non-trivalent components from a graph. Furthermore, for γ ∈ B̂′ and γ′ ∈ B̂, we have

⟨γ, \frac{ℓ}{2}γ′⟩ = ⟨∂γ, γ′⟩ . (27)

For γ, γ′ ∈ B̂′, we have the following (combinatorial) formula:

(28)  ⟨exp(∂)(γγ′), 1⟩ = ⟨exp(∂)γ, exp(∂)γ′⟩ .

Proof. The formula (27) should be clear from the definitions. Let us investigate (28) a bit more. We can assume that γ and γ′ are Jacobi diagrams with l resp. l′ univalent vertices and l + l′ = 2n with n ∈ ℕ_{0}. So we have to prove

\frac{∂^{n}}{n!} (γγ′) = ∑_{\begin{matrix}m,m′=0 \\ l−2m=l′−2m′ \end{matrix}}^{∞} ⟨\frac{∂^{m}}{m!} γ, \frac{∂^{m′}}{m′!} γ′⟩ ,

since ⟨·, 1⟩ : B̂ → ^{t}B̂ means to remove the components with at least one univalent vertex. Recalling the meaning of ⟨·, ·⟩, it should be clear that (28) follows from the fact that applying \frac{∂^{k}}{k!} on a Jacobi diagram means to glue all subsets of 2k of its univalent vertices to k pairs in all possible ways. □

3.3. An 𝔰𝔩_{2}-action on the space of graph homology. In this short section we want to extend the space of graph homology slightly. This is mainly due to two reasons: When we defined the expression Γ̂(Γ) for two Jacobi diagrams Γ and Γ′, we restricted ourselves to the case that Γ or Γ′ does not contain a component with an ℓ. Secondly, we have not given the zero-wheel w_{0} a meaning yet. We do this by adding an element ◯ to the various spaces of graph homology.

Definition 8. The extended space of graph homology is the space B̂[[◯]]. Further, we set w_{0} := ◯, which, at least pictorially, is in accordance with the definition of w_{k} for k > 0.

Note that this element is not depicting a Jacobi diagram as we have defined it. Nevertheless, we want to use the notion that ◯ has no univalent and no trivalent vertices, i.e. the homogeneous component of degree zero of B̂[[◯]] is ℚ[[◯]]. When defining Γ/(u, u′) for a Jacobi diagram Γ with two univalent vertices u and u′, i.e. gluing u to u′, we assumed that u and u′ are not the vertices of one component ℓ of Γ. Now we extend this definition by defining Γ/(u, u′) to be the extended graph homology class we get by replacing ℓ with ◯, whenever u and u′ are the two univalent vertices of a component ℓ of Γ. Doing so, we can give the expression γ̂(γ′) ∈ B̂[[◯]] a meaning with no restrictions on the two graph homology classes γ, γ′ ∈ B̂, i.e. every γ ∈ B̂[[◯]] defines a ^{t}B̂[[◯]]-linear map

(29)  γ̂ : B̂[[◯]] → B̂[[◯]].

Example 4. We have

(30)  ∂ℓ = ◯.

Remark 5. We can similarly extend ⟨·, ·⟩ : B̂′ × B̂ → ^{t}B̂ to a ^{t}B̂[[◯]]-bilinear form

(31)  ⟨·, ·⟩ : B̂[[◯]] × B̂[[◯]] → ^{t}B̂[[◯]].

Both ℓ/2 and ∂ are two operators acting on the extended space of graph homology, the first one just multiplication with ℓ/2. By calculating their commutator, we show that they induce a natural structure of an 𝔰𝔩_{2}-module on B̂[[◯]].

Proposition 5. Let H : B̂[[◯]] → B̂[[◯]] be the linear operator which acts on γ ∈ B̂_{k,l}[[◯]] by

(32)  Hγ = (\frac{1}{2} ◯ +l) γ.

We have the following commutator relations in End B̂[[◯]]:

(33)  [ℓ/2, ∂] = −H,
(34)  [H, ℓ/2] = 2 · ℓ/2,

and

(35)  [H, ∂] = −2∂,

i.e. the triple (ℓ/2, −∂, H) defines a 𝔰𝔩_{2}-operation on B̂[[◯]].

Proof. Equations (34) and (35) follow from the fact that multiplying by ◯ commutes with ℓ/2 and ∂, and from the fact that ℓ/2 is an operator of degree 2 with respect to the grading given by the number of univalent vertices, whereas ∂ is an operator of degree −2 with respect to the same grading. It remains to look at (33). For γ ∈ B̂_{k,l}[[◯]], we calculate

[ℓ, ∂]γ = ℓ∂(γ) − ∂(ℓγ) = ℓ∂(γ) − ∂(ℓ)γ − ℓ∂(γ) − ∂(ℓ, γ) = − ◯ γ − 2lγ = −2Hγ. (36)
□

Remark 6. Since B̂[[◯]] is infinite-dimensional, we have unfortunately difficulties to apply the standard theory of 𝔰𝔩_{2}-representations to this 𝔰𝔩_{2}-module. For example, there are no eigenvectors for the operator H.

3.4. Closed and connected graphs, the closure of a graph. As the number of connected components of a Jacobi diagram is preserved by the IHXand AS-relations each graph homology space inherits a grading by the number of connected components. For any k ∈ ℕ_{0} we define B^{k} to be the subspace of B spanned by all Jacobi diagrams with exactly k connected components. Similarly, we define ^{t}B^{k}, B̂^{k}, ^{t}B̂^{k}. We have B = ⨁_{k=0}^{∞} B^{k} with B^{0} = ℚ · 1. Analogous results hold for ^{t}B, B̂, ^{t}B̂.

Definition 9. A graph homology class γ is called closed if γ ∈ ^{t}B̂. The class γ is called connected if γ ∈ B̂^{1}. The connected component of γ is defined to be pr^{1}(γ) where pr^{1} : B̂ = ∏_{i=0}^{∞} B̂^{i} → B̂^{1} is the canonical projection. The closure ⟨γ⟩ of γ is defined by ⟨γ⟩ := ⟨γ, exp(ℓ/2)⟩. The connected closure ⟨⟨γ⟩⟩ of γ is defined to be the connected component of the closure ⟨γ⟩ of γ.

For every finite set L, we define P_{2}(L) to be the set of partitions of L into subsets of two elements. With this definition, we can express the closure of a Jacobi diagram Γ as

⟨Γ⟩ = ∑_{π∈P_{2}(L)} Γ/π. (37)

Example 5. We have ⟨w_{2}⟩ = �, ⟨⟨w_{2}⟩⟩ = �, ⟨w_{2}^{2}⟩ = 2�_{2} + �^{2}, ⟨⟨w_{2}^{2}⟩⟩ = 2�_{2}.

Let L_{1}, …, L_{n} be finite and pairwise disjoint sets. We set L := ⨆_{i=1}^{n} L_{i}. Let π ∈ P_{2}(L) be a partition of L in 2-element-subsets. We say that a pair l, l′ ∈ L is linked by π if there is an i ∈ {1, … n} such that l, l′ ∈ L_{i} or {l, l′} ∈ π. We say that π connects the sets L_{1}, …, L_{n} if and only if for each pair l, l′ ∈ L there is a chain of elements l_{1}, …, l_{k} such that l is linked to l_{1}, l_{i} is linked to l_{i+1} for i ∈ {1, …, k − 1} and l_{k} is linked to l′. The subset of P_{2}(L) of partitions π connecting L_{1}, …, L_{n} is denoted by P_{2}({L_{1}, …, L_{n}}). We have

P_{2}(L) = ⨆ ; ⨆ ℑ={1,...,n} { ⨆_{I∈ℑ} π_{I} : π_{I} ∈ P_{2}({L_{i} : i ∈ I})} . (38)

Here, ⨆ ℑ = {1, …, n} means that ℑ is a partition of {1, …, n} in disjoint subsets. Let Γ_{1}, …, Γ_{n} be connected Jacobi diagrams. We denote by Γ := ∏_{i=1}^{n} Γ_{i} the product over all these Jacobi diagrams. Let L_{i} be the set of legs of Γ_{i} and denote by L := ⨆_{i=1}^{n} L_{i} the set of all legs of Γ. For every partition π ∈ P_{2}(L) the graph Γ/π is connected if and only if π ∈ P_{2}({L_{1}, …, L_{n}}). Using (38) we have

(39) ⟨Γ⟩ = ∑_{π∈P_{2}(L)} Γ/π = ∑ ; ⨆ ℑ={1,...,n} ∏_{I∈ℑ} ∑_{π∈P_{2}({L_{i}:i∈I})} (∏_{i∈I} Γ_{i}) /π

= ∑ ; ⨆ ℑ={1,...,n} ∏_{I∈ℑ} ⟨⟨∏_{i∈I} Γ_{i}⟩⟩ .

With this result we can prove the following Proposition:

Proposition 6. For any connected graph homology class γ we have

(40) exp ⟨⟨exp γ⟩⟩ = ⟨exp γ⟩ .

Note that both sides are well-defined in B̂ since γ and ⟨⟨⋯ ⟩⟩ as connected graphs have no component in degree zero.

Proof. Let Γ be any connected Jacobi diagram. By (39) we have

⟨Γ^{n}⟩ = ∑ ; ⨆ ℑ=\{1,...,n\} ∏_{I∈ℑ} ⟨⟨Γ^{#I}⟩⟩ = ∑_{λ∈P(n)} n! ∏_{i=1}^{∞} \frac{1}{λ_{i}!} (⟨⟨Γ^{i}⟩⟩ /i!)^{λ_{i}} .

By linearity this result holds also if we substitute Γ by the connected graph homology class γ. Using this,

⟨exp γ⟩ = ∑_{n=0}^{∞} \frac{1}{n!} ⟨γ^{n}⟩ = ∑_{n=0}^{∞} \frac{1}{n!} ∑_{λ∈P(n)} n! ∏_{i=1}^{∞} \frac{1}{λ_{i}!} (⟨⟨γ^{i}⟩⟩ /i!)^{λ_{i}}

= ∏_{i=1}^{∞} ∑_{λ=0}^{∞} \frac{1}{λ!} (⟨⟨γ^{i}⟩⟩ /i!)^{λ} = ∏_{i=1}^{∞} exp (⟨⟨γ^{i}⟩⟩ /i!) = exp ⟨⟨exp γ⟩⟩ .

□

3.5. Polywheels.

Definition 10. For each n ∈ ℕ_{0} we set w̃_{2n} := −w_{2n}. Let λ be a partition of n. We set

w̃_{2λ} := ∏_{i=1}^{∞} w̃_{2i}^{λ_{i}}. (41)

The closure \left⟨{ \atop w̃_{2λ}}\right⟩ of w̃_{2λ} is called a polywheel. The subspace in ^{t}B spanned by all polywheels is denoted by W and called the polywheel subspace. The subalgebra in ^{t}B spanned by all polywheels is denoted by C and called the algebra of polywheels. The connected closure ⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩ of w̃_{2λ} is called a connected polywheel.

Remark 7. As discussed by J. Sawon in his thesis [16], W is proper graded subspace of ^{t}B. From degree eight on, ^{t}B_{k} is considerably larger than W_{k}. On the other hand it is unknown (at least to the author) if the inclusion C ⊆ ^{t}B is proper.

Remark 8. The subalgebra C′ in^{t}B spanned by all connected polywheels equals C. This is since we can use (40) to express every polywheel as a polynomial of connected polywheels and vice versa.

Example 6. Using Proposition 6 we calculated the following expansions of the connected polywheels in terms of wheels:

\begin{aligned}⟨\left⟨{ \atop w̃_{2}}\right⟩⟩ & = \left⟨{ \atop w̃_{2}}\right⟩ \\ ⟨⟨w̃_{2}^{2}⟩⟩ & = ⟨w̃_{2}^{2}⟩ − \left⟨{ \atop w̃_{2}}\right⟩^{2} \\ ⟨\left⟨{ \atop w̃_{4}}\right⟩⟩ & = \left⟨{ \atop w̃_{4}}\right⟩ \\ ⟨⟨w̃_{2}^{3}⟩⟩ & = ⟨w̃_{2}^{3}⟩ − 3 \left⟨{ \atop w̃_{2}}\right⟩ ⟨w̃_{2}^{2}⟩ + 2 \left⟨{ \atop w̃_{2}}\right⟩^{3} \\ ⟨⟨w̃_{2}w̃_{4}⟩⟩ & = ⟨w̃_{2}w̃_{4}⟩ − \left⟨{ \atop w̃_{2}}\right⟩ \left⟨{ \atop w̃_{4}}\right⟩ \\ ⟨\left⟨{ \atop w̃_{6}}\right⟩⟩ & = \left⟨{ \atop w̃_{6}}\right⟩ \\ ⟨⟨w̃_{2}^{4}⟩⟩ & = ⟨w̃_{2}^{4}⟩ − 4 \left⟨{ \atop w̃_{2}}\right⟩ ⟨w̃_{2}^{3}⟩ − 3 ⟨w̃_{2}^{2}⟩^{2} + 12 \left⟨{ \atop w̃_{2}}\right⟩^{2} − 6 \left⟨{ \atop w̃_{2}}\right⟩^{4} \\ ⟨⟨w̃_{2}^{2}w̃_{4}⟩⟩ & = ⟨w̃_{2}^{2}w̃_{4}⟩ − 2 \left⟨{ \atop w̃_{2}}\right⟩ ⟨w̃_{2}w̃_{4}⟩ − ⟨w̃_{2}^{2}⟩ \left⟨{ \atop w̃_{4}}\right⟩ + 2 \left⟨{ \atop w̃_{2}}\right⟩^{2} \left⟨{ \atop w̃_{4}}\right⟩ \\ ⟨⟨w̃_{2}w̃_{6}⟩⟩ & = ⟨w̃_{2}w̃_{6}⟩ − \left⟨{ \atop w̃_{2}}\right⟩ \left⟨{ \atop w̃_{6}}\right⟩ \\ ⟨⟨w̃_{4}^{2}⟩⟩ & = ⟨w̃_{4}^{2}⟩ − \left⟨{ \atop w̃_{4}}\right⟩^{2} \\ ⟨\left⟨{ \atop w̃_{8}}\right⟩⟩ & = \left⟨{ \atop w̃_{8}}\right⟩ . \end{aligned} (42)

4. Holomorphic symplectic manifolds

4.1. Definition and general properties.

Definition 11. A holomorphic symplectic manifold (X, σ) is a compact complex manifold X together with an everywhere non-degenerate holomorphic twoform σ ∈ H^{0}(X, Ω_{X}^{2}). Here, we call σ everywhere non-degenerate if σ induces an isomorphism T_{X} → Ω_{X}.

The holomorphic symplectic manifold (X, σ) is called irreducible if it is simplyconnected and H^{0}(X, Ω_{X}^{2}) is one-dimensional, i.e. spanned by σ.

It follows immediately that every holomorphic symplectic manifold X has trivial canonical bundle whose sections are multiples of σ^{n}, and, therefore, vanishing first Chern class. In fact, all odd Chern classes vanish:

Proposition 7. Let X be a complex manifold and E a complex vector bundle on X. If E admits a symplectic two-form, i.e. there exists a section σ ∈ H^{0}(X, Λ^{2}E^{∗}) such that the induced morphism E → E^{∗} is an isomorphism, all odd Chern classes of E vanish.

Remark 9. That the odd Chern classes of E vanish up to two-torsion follows immediately from the fact c_{2k+1}(E) = −c_{2k+1}(E^{∗}) for k ∈ ℕ_{0}.

The following proof using the splitting principle has been suggested to me by Manfred Lehn.

Proof. We prove the proposition by induction over the rank of E. For rk E = 0, the claim is obvious. By the splitting principle (see e.g. [5]), we can assume that E has a subbundle L of rank one. Let L^{⊥} be the σ-orthogonal subbundle to L of E. Since σ is symplectic, L^{⊥} is of rank n − 1 and L is a subbundle of L^{⊥}. We have the following short exact sequences of bundles on X:

0 ⟶ L ⟶ E ⟶ E/L ⟶ 0

and

0 ⟶ L^{⊥}/L ⟶ E/L ⟶ E/L^{⊥} ⟶ 0.

Since σ induces a symplectic form on L^{⊥}/L, by induction, all odd Chern classes of this bundle of rank rk E − 2 vanish. Furthermore, note that σ induces an isomorphism between L and (E/L^{⊥})^{∗}, so all odd Chern classes of L ⊕ E/L^{⊥} vanish. Now, the two exact sequences give us c(E) = c(L⊕E/L^{⊥})·c(L^{⊥}/L). Therefore, we can conclude that all odd Chern classes of E vanish. □

Proposition 8. For any irreducible holomorphic symplectic manifold (X, σ) of dimension 2n and k ∈ 0, …, n the space H^{2k}(X, O_{X}) is one-dimensional and spanned by the cohomology class [σ̄]^{k}.

Proof. See [2]. □

4.2. A pairing on the cohomology of a holomorphic symplectic manifold. Let (X, σ) be a holomorphic symplectic manifold. There is a natural pairing of coherent sheafs

Λ_{∗}T_{X} ⊗ Λ^{∗}Ω_{X} → O_{X}. (43)

As the natural morphism from Λ_{∗}T_{X} to Λ^{∗}T_{X} is an isomorphism and Λ^{∗}T_{X} can be identified with Λ^{∗}Ω_{X} by means of the symplectic form, we therefore have a natural map

Λ^{∗}Ω ⊗ Λ^{∗}Ω_{X} → O_{X}. (44)

We write

⟨·, ·⟩ : H^{p}(X, Ω^{∗}) ⊗ H^{q}(X, Ω^{∗}) → H^{p+q}(X, O_{X}), (α, β) ↦ ⟨α, β⟩ (45)

for the induced map for any p, q ∈ ℕ_{0}. In [10] we proved the following proposition:

Proposition 9. For any α ∈ H^{∗}(X, Ω^{∗}) we have

(46) ∫_{X} α exp σ = ∫_{X} ⟨α, exp σ⟩ exp σ.

4.3. Example series. There are two main series of examples of irreducible holomorphic symplectic manifolds. Both of them are based on the Hilbert schemes of points on a surface: Let X be any smooth projective surface over ℂ and n ∈ ℕ_{0}. By X^{[n]} we denote the Hilbert scheme of zero-dimensional subschemes of length n of X. By a result of Fogarty ([4]), X^{[n]} is a smooth projective variety of dimension 2n. The Hilbert scheme can be viewed as a resolution ρ : X^{[n]} → X^{(n)} of the n-fold symmetric product X^{(n)} := X^{n}/𝔖_{n}. The morphism ρ, sending closed points, i.e. subspaces of X, to their support counting multiplicities, is called the Hilbert-Chow morphism. Let α ∈ H^{2}(X, ℂ) be any class. The class ∑_{i=1}^{n} pr_{i}^{∗} α ∈ H^{2}(X^{n}, ℂ) is invariant under the action of 𝔖_{n}, where pr_{i} : X^{n} → X denotes the projection on the i^{th} factor. Therefore, there exists a class α^{(n)} ∈ H^{2}(X^{(n)}, ℂ) with π^{∗}α^{(n)} = ∑_{i=1}^{n} pr_{i}^{∗} α, where π : X^{n} → X^{(n)} is the canonical projection. Using ρ this induces a class α^{[n]} in H^{2}(X^{[n]}, ℂ). If X is a K3 surface or an abelian surface, there exists a holomorphic symplectic form σ ∈ H^{2,0}(X) ⊆ H^{2}(X, ℂ). It was shown by Beauville in [2] that σ^{[n]} is again symplectic, so (X^{[n]}, σ^{[n]}) is a holomorphic symplectic manifold.

Example 7. For any K3 surface X and holomorphic symplectic form σ ∈ H^{2,0}(X), the pair (X^{[n]}, σ^{[n]}) is in fact an irreducible holomorphic symplectic manifold.

This has also been proven by Beauville. In the case of an abelian surface A, we have to work a little bit more as A^{[n]} is not irreducible in this case: Let A be an abelian surface and let us denote by s : A^{[n]} → A the composition of the summation morphism A^{(n)} → A with the Hilbert-Chow morphism ρ : A^{[n]} → A^{(n)}.

Definition 12. For any n ∈ ℕ, the n^{th} generalised Kummer variety A^{[[n]]} is the fibre of s over 0 ∈ A. For any class α ∈ H^{2}(A, ℂ), we set α^{[[n]]} := α^{[n]}|_{A^{^{[}^{[n]}^{]}}}.

Remark 10. For n = 2 the generalised Kummer variety coincides with the Kummer model of a K3 surface (therefore the name).

Example 8. For every abelian surface A and holomorphic symplectic form σ ∈ H^{2,0}(A), the pair (A^{[[n]]}, σ^{[[n]]}) is an irreducible holomorphic symplectic manifold of dimension 2n − 2.

The proof can also be found in [2].

4.4. About α^{[n]} and α^{[[n]]}. Let X be any smooth projective surface and n ∈ ℕ_{0}. Let X^{[n,n+1]} denote the incidence variety of all pairs (ξ, ξ′) ∈ X^{[n]}×X^{[n+1]} with ξ ⊆ ξ′ (see [3]). We denote by ψ : X^{[n,n+1]} → X^{[n+1]} and by ϕ : X^{[n,n+1]} → X^{[n]} the canonical maps. There is a third canonical map χ : X^{[n,n+1]} → X mapping (ξ, ξ′) ↦ x if ξ′ is obtained by extending ξ at the closed point x ∈ X.

Proposition 10. For any α ∈ H^{2}(X, ℂ) we have

ψ^{∗}α^{[n+1]} = ϕ^{∗}α^{[n]} + χ^{∗}α. (47)

Proof. Let p : X^{(n)} × X → X^{(n)} and q : X^{(n)} × X → X denote the canonical projections. Let τ : X^{(n)} × X → X^{(n+1)} the obvious symmetrising map. The following diagram

X^{[n,n+1]}  X^{[n,n+1]}
(ϕ,χ)↓  ↓^{ψ}
X^{[n]} × X  X^{[n+1]}
ρ×id_{X} ↓  ↓^{ρ′}
X^{(n)} × X ⟶^{τ} X^{(n+1)}
π×id_{X} ↑  ↑_{π′}
X^{n+1}  X^{n+1}

is commutative. (Note that we have primed some maps to avoid name clashes.) We claim that τ^{∗}α^{(n+1)} = p^{∗}α^{(n)} + q^{∗}α. In fact, since

(π × id_{X})^{∗}τ^{∗}α^{(n+1)} = π^{′∗}α^{(n+1)} = ∑_{i=1}^{n+1} pr_{i}^{∗} α,

this follows from the definition of α^{(n)}. Finally, we can read off the diagram that

ψ^{∗}α^{[n+1]} = ψ^{∗}ρ^{′∗}α^{(n+1)} = (ϕ, χ)^{∗}(ρ × id_{X})^{∗}τ^{∗}α^{(n+1)}

= (ϕ, χ)^{∗}(ρ × id_{X})^{∗}(p^{∗}α^{(n)} + q^{∗}α) = ϕ^{∗}α^{[n]} + χ^{∗}α.

□

Proposition 11. Let X = X_{1} ⊔ X_{2} be the disjoint union of two projective smooth surfaces X_{1} and X_{2}. We then have

X^{[n]} = ⨆_{n_{1}+n_{2}=n} X_{1}^{[n_{1}]} × X_{2}^{[n_{2}]}. (48)

If α ∈ H^{2}(X, ℂ) decomposes as α|_{X_{1}} = α_{1} and α|_{X_{2}} = α_{2}, then α^{[n]} decomposes as

α^{[n]}|_{X_{!}^{[n_{1}]}×X_{2}^{[n_{2}]}} = pr_{1}^{∗} α_{1}^{[n_{1}]} + pr_{2}^{∗} α_{2}^{[n_{2}]}. (49)

Proof. The splitting of X^{[n]} follows from the universal property of the Hilbert scheme and is a well-known fact. The statement on α^{[n]} is easy to prove and so we shall only give a sketch: Let us denote by i : X_{1}^{[n_{1}]} × X_{2}^{[n_{2}]} → X^{[n]} the natural inclusion. Furthermore let j : X_{1}^{(n_{1})} ×X_{2}^{(n_{2})} → X^{(n)} denote the natural symmetrising map. The following diagram is commutative:

\begin{matrix}X_{1}^{[n_{1}]} × X_{2}^{[n_{2}]} & ⟶^{i} & X^{[n]} \\ ^{ρ_{1}×ρ_{2}}↓ & & ↓^{ρ} \\ X_{1}^{(n_{1})} × X_{2}^{(n_{2})} & ⟶_{j} & X^{(n)}, \end{matrix} (50)

where the ρ_{i} : X_{i}^{[n_{i}]} → X_{i}^{(n_{i})} are the Hilbert-Chow morphisms. Since j^{∗}α^{(n)} = pr_{1}^{∗} α_{1}^{(n_{1})} + pr_{2}^{∗} α_{2}^{(n_{2})}, the commutativity of the diagram proves the statement on α^{[n]}. □

Let A be again an abelian surface and n ∈ ℕ. Since A acts on itself by translation, there is also an induced operation of A on the Hilbert scheme A^{[n]}. Let us denote the restriction of this operation to the generalised Kummer variety A^{[[n]]} by ν : A×A^{[[n]]} → A^{[n]}. It fits into the following cartesian square:

\begin{matrix}A × A^{[[n]]} & ⟶^{ν} & A^{[n]} \\ ^{pr}^{1}↓ & & ↓^{s} \\ A & ⟶_{n} & A, \end{matrix} (51)

where s is the summation map as having been defined above and n : A → A, a ↦ na is the (multiplication-by-n)-morphism. Since n is a Galois cover of degree n^{4}, the same holds true for ν.

Proposition 12. For any α ∈ H^{2}(A, ℂ), we have

ν^{∗}α^{[n]} = n pr_{1}^{∗} α + pr_{2}^{∗} α^{[[n]]}. (52)

Proof. By the Künneth decomposition theorem, we know that ν^{∗}α^{[n]} splits:

ν^{∗}α^{[n]} = pr_{1}^{∗} α_{1} + pr_{2}^{∗} α_{2}.

Set ι_{1} : A → A×A^{[[n]]}, a ↦ (a, ξ_{0}) and ι_{2} : A^{[[n]]} → A×A^{[[n]]}, ξ ↦ (0, ξ), where ξ_{0} is any subscheme of length n concentrated in 0. We have

α_{1} = ι_{1}^{∗}ν^{∗}α^{[n]} = (ρ ∘ ν ∘ ι_{1})^{∗}α^{(n)} = (a ↦ (\underbrace{a, … a}_{n}))^{∗}α^{(n)} = nα (53)

and

α_{2} = ι_{2}^{∗}ν^{∗}α^{[n]} = i^{∗}α^{[n]} = α^{[[n]]}, (54)

where i : A^{[[n]]} → A^{[n]} is the natural inclusion map, thus proving the proposition. □

4.5. Complex genera of Hilbert schemes of points on surfaces. The following theorem is an adaption of Theorem 4.1 of [3] to our context.

Theorem 1. Let P be a polynomial in the variables c_{1}, c_{2}, … and α over ℚ. There exists a polynomial P̃ ∈ ℚ[z_{1}, z_{2}, z_{3}, z_{4}] such that for every smooth projective surface X, α ∈ H^{2}(X, ℚ) and n ∈ ℕ_{0} we have:

∫_{X^{[n]}} P (c_{∗}(X^{[n]}), α^{[n]}) = P̃ (∫_{X} α^{2}/2, ∫_{X} c_{1}(X)α, ∫_{X} c_{1}(X)^{2}/2, ∫_{X} c_{2}(X)) . (55)

Proof. The proof goes along the very same lines as the proof of Proposition 0.5 in [3] (see there). The only new thing we need is Proposition 10 of this paper to be used in the induction step of the adapted proof of Proposition 3.1 of [3] to our situation. □

Let R be any ℚ-algebra (commutative and with unit) and let ϕ ∈ R[[c_{1}, c_{2}, …]] be a non-vanishing power series in the universal Chern classes such that ϕ is multiplicative with respect to the Whitney sum of vector bundles, i.e.

ϕ(E ⊕ F) = ϕ(E)ϕ(F) (56)

for all complex manifolds and complex vector bundles E and F on X. Any ϕ with this property induces a complex genus, also denoted by ϕ, by setting ϕ(X) := ∫_{X} ϕ(T_{X}) for X a compact complex manifold. Let us call such a ϕ multiplicative.

Remark 11. By Hirzebruch’s theory of multiplicative sequences and complex genera ([6]), we know that

(1) each complex genus is induced by a unique multiplicative ϕ, and

(2) the multiplicative elements in R[[c_{1}, c_{2}, …]] are exactly those of the form exp(∑_{k=1}^{∞} a_{k}s_{k}) with a_{k} ∈ R.

More or less formally the following theorem follows from Theorem 1.

Theorem 2. For each multiplicative ϕ ∈ R[[c_{1}, c_{2}, …]], there exist unique power series A_{ϕ}(p), B_{ϕ}(p), C_{ϕ}(p), D_{ϕ}(p) ∈ pR[[p]] with vanishing constant coefficient such that for all smooth projective surfaces X and α ∈ H^{2}(X, ℂ) we have:

(57) ∑_{n=0}^{∞} (∫_{X^{[n]}} ϕ(X^{[n]}) exp(α^{[n]})) p^{n}

= exp (A_{ϕ}(p) ∫_{X} α^{2}/2 + B_{ϕ}(p) ∫_{X} c_{1}(X)α

+ C_{ϕ}(p) ∫_{X} c_{1}^{2}(X)/2 + D_{ϕ}(p) ∫_{X} c_{2}(X)) .

The first terms of A_{ϕ}(p), B_{ϕ}(p), C_{ϕ}(p), D_{ϕ}(p) are given by

(58) A_{ϕ}(p) = p + O(p^{2}), B_{ϕ}(p) = ϕ_{1}p + O(p^{2}), C_{ϕ}(p) = ϕ_{11}p + O(p^{2}), and

D_{ϕ}(p) = ϕ_{2}p + O(p^{2}),

where ϕ_{1} is the coefficient of c_{1} in ϕ, ϕ_{11} the coefficient of c_{1}^{2}/2 and ϕ_{2} the coefficient of c_{2}.

Proof. This theorem is again an adaption of a theorem (Theorem 4.2) of [3] to our context. Nevertheless, let us give the proof here: Set K := {(X, α) : X is a smooth projective surface and α ∈ H^{2}(X, ℂ)} and let γ : K → ℚ^{4} be the map (X, α) ↦ (α^{2}/2, c_{1}(X)α, c_{1}(X)^{2}/2, c_{2}(X)). Here, we have supressed the integral signs ∫_{X} and interpret the expressions α^{2}, etc. as intersection numbers on X. The image of K spans the whole ℚ^{4} (for explicit generators, we refer to [3]). Now let us assume that a (X, α) ∈ K decomposes as (X, α) = (X_{1}, α_{1}) ⊔ (X_{2}, α_{2}). By the multiplicative behaviour of ϕ and exp we see that

∫_{X^{[n]}} ϕ(c_{∗}(X^{[n]})) exp(α^{[n]})

= ∑_{n_{1}+n_{2}=n} (∫_{X_{1}^{[n_{1}]}} ϕ(c_{∗}(X^{[n_{1}]})) exp(α_{2}^{[n_{1}]})) (∫_{X_{2}^{[n_{2}]}} ϕ(c_{∗}(X^{[n_{2}]})) exp(α_{2}^{[n_{2}]})) ,

whereas H_{ϕ}(p)(X, α) := ∑_{n=0}^{∞} (∫_{X^{[n]}} ϕ(X^{[n]}) exp(α^{[n]})) p^{n} fulfills

H_{ϕ}(p)(X, α) = H_{ϕ}(p)(X_{1}, α_{1})H_{ϕ}(p)(X_{2}, α_{2}). (∗)

Since H_{ϕ}(p) : K → ℚ^{4} factors through γ and a map h : ℚ^{4} → R[[p]] by Theorem 1 and as the image of γ is Zariski dense in ℚ^{4}, we conclude from (∗) that log h is a linear function which proves the first part of the theorem. To get the first terms of the power series, we expand both sides of (57). The left hand side expands as

1 + (α^{2}/2 + ϕ_{1}c_{1}(X)α + \frac{ϕ_{11}}{2} c_{1}^{2}(X) + ϕ_{2}c_{2}(X))p + O(p^{2}), (59)

while the right hand side expands as

1 + (A_{1}α^{2}/2 + B_{1}c_{1}(X)α + C_{1}c_{1}(X)^{2} + D_{1}c_{2}(X))p + O(p^{2}), (60) where A_{1}, B_{1}, C_{1}, D_{1} are the linear coefficients of A_{ϕ}, B_{ϕ}, C_{ϕ}, and D_{ϕ}, which can therefore be read off by comparing the expansions. □

Corollary 1. Let X be any smooth projective surface, α ∈ H^{2}(X, ℂ), and n ∈ ℕ_{0}. Then

∫_{X^{[n]}} exp(α^{[n]} + ᾱ^{[n]}) = \frac{1}{n!} (∫_{X} αᾱ)^{n} . (61)

For X = A an abelian surface and n ∈ ℕ, we get

∫_{A^{^{[}^{[n]}^{]}}} exp(α^{[[n]]} + ᾱ^{[[n]]}) = \frac{n}{(n − 1)!} (∫_{X} αᾱ)^{n−1} . (62)

Proof. By Theorem 2, in ℂ[[q]]:

∑_{n=0}^{∞} (∫_{X^{[n]}} exp(q^{\frac{1}{2}} (α^{[n]} + ᾱ^{[n]}))) p^{n} = exp(pq ∫_{X} αᾱ + O(p^{2})), (63)

which proves the first part of the corollary by comparing coefficients of q. For the Kummer case, we calculate

∫_{A^{^{[}^{[n]}^{]}}} exp(α^{[[n]]} + ᾱ^{[[n]]}) = \frac{∫_{A^{^{[}^{[n]}^{]}}} exp(α^{[[n]]} + ᾱ^{[[n]]}) ∫_{A} exp(nα + nᾱ)}{∫_{A} exp(nα + nᾱ)}

= n^{2} \frac{∫_{A^{[n]}} exp(α^{[n]} + ᾱ^{[n]})}{∫_{A} exp(α + ᾱ)},

which proves the rest of the corollary. □

Let ch be the universal Chern character. By s_{k} = (2k)!ch_{2k} we denote its components. They span the whole algebra of characteristic classes, i.e. we have ℚ[s_{1}, s_{2}, …] = ℚ[c_{1}, c_{2}, …]. Let us fix the power series

ϕ := exp(∑_{k=1}^{∞} a_{2k}s_{2k}t^{k}) ∈ ℚ[a_{2}, a_{4}, …][t][[c_{1}, c_{2}, …]].

This multiplicative series gives rise to four power series

A_{ϕ}(p), B_{ϕ}(p), C_{ϕ}(p), D_{ϕ}(p) ∈ pR[[p]]

according to the previous Theorem 2. We shall set for the rest of this article

A(t) := A_{ϕ}(1), and D(t) := D_{ϕ}(1) (64)

The constant terms of these power series in t are given by

A(t) = 1 + O(t), and D(t) = O(t). (65)

5. Rozansky-Witten classes and invariants

The idea to associate to every graph Γ and every hyperkähler manifold X a cohomology class RW_{X}(Γ) is due to L. Rozansky and E. Witten (c.f. [15]). M. Kapranov showed in [8] that the metric structure of a hyperkähler manifold is not nessessary to define these classes. It was his idea to build the whole theory upon the Atiyah class and the symplectic structure of an irreducible holomorphic symplectic manifold. We will make use of his definition of Rozansky-Witten classes in this section. A very detailed text on defining Rozansky-Witten invariants is the thesis by J. Sawon [16].

5.1. Definition. Let (X, σ) be a holomorphic symplectic manifold. Let us work in the category of complexes of coherent sheaves on X. In this category, we have for every n ∈ ℤ a functor V ↦ V [n] that shifts a complex V by n to the left. Due to the Koszul sign rule (i.e. the natural map (V [m]) ⊗ (W [n]) → (W [n]) ⊗ (V [m]) for sheaves V and W and integers n and m incorporates a sign (−1)^{mn}), we have S^{n}(V [1]) = (Λ^{n}V)[n] and S_{n}(V [1]) = (Λ_{n}V)[n]. Every Jacobi diagram Γ with k trivalent and l univalent vertices defines in the category of complexes of coherent sheaves on X a morphism

Φ^{Γ} : S_{k}Λ_{3}(T_{X}[−1]) ⊗ S_{l}(T_{X}[−1]) → S^{e}S^{2}(T_{X}[−1]), (66)

where T_{X}[−1] is the tangent sheaf of X shifted by one and 2e = 3k + l. By the sign rule above, this is equivalent to being given a map:

(Λ_{k}S_{3}T_{X} ⊗ Λ_{l}T_{X})[−3k − l] → (S^{e}Λ^{2}T_{X})[−2e], (67)

which is induced by a map

Λ_{k}S_{3}T_{X} ⊗ Λ_{l}T_{X} → S^{e}Λ^{2}T_{X} (68)

in the category of coherent sheaves on X. This gives rise to a map

(69) Ψ^{Γ} : Λ_{k}S_{3}T_{X} ⊗ S_{e}Λ_{2}Ω_{X} → Λ^{l}Ω_{X}.

Let α̃ ∈ H^{1}(X, Ω ⊗ End T_{X}) be the Atiyah class of X, i.e. α̃ represents the extension class of the sequence

0 ⟶ Ω_{X} ⊗ T_{X} ⟶ J^{1}T_{X} ⟶ T_{X} ⟶ 0 (70)

in Ext_{X}^{1}(T_{X}, Ω_{X} ⊗ T_{X}) = H^{1}(X, Ω_{X} ⊗ End T_{X}). Here, J^{1}T_{X} is the bundle of one-jets of sections of T_{X} (for more on this, see [8]). The Atiyah class can also be viewed as the obstruction for a global holomorphic connection to exist on T_{X}. We set α := i/(2π)α̃. We use σ to identify the tangent bundle T_{X} of X with its cotangent bundle Ω_{X}. Doing this, α can be viewed as an element of H^{1}(X, T_{X}^{⊗3}). Now the point is that α is not any such element. The following proposition was proven by Kapranov in [8]:

Proposition 13.

α ∈ H^{1}(X, S_{3}T_{X}) ⊆ H^{1}(X, T_{X}^{⊗3}). (71)

Therefore, α^{∪k} ∪ σ^{∪l} ∈ H^{k}(X, Λ_{k}S_{3}T_{X} ⊗ S_{l}Λ_{2}Ω_{X}). Applying the map Ψ^{Γ} on the level of cohomology eventually leads to an element

RW_{X,σ}(Γ) := Ψ_{∗}^{Γ}(α^{∪k} ∪ σ^{∪l}) ∈ H^{k}(X, Ω_{X}^{l}). (72)

We call RW_{X,σ}(Γ) the Rozansky-Witten class of (X, σ) associated to Γ. For a ℂ-linear combination γ of Jacobi diagrams, RW_{X,σ}(γ) is defined by linear extension. In [8], Kapranov also showed the following proposition, which is crucial for the next definition. It follows from a Bianchi-identity for the Atiyah class.

Proposition 14. If γ is a ℚ-linear combination of Jacobi diagrams that is zero modulo the anti-symmetry and IHX relations, then RW_{X,σ}(γ) = 0.

Definition 13. We define a double-graded linear map

RW_{X,σ} : B̂ → H^{∗}(X, Ω_{X}^{∗}), (73)

which maps B_{k,l} into H^{k}(X, Ω_{X}^{l}) by mapping a homology class of a Jacobi diagram Γ to RW_{X,σ}(Γ).

Definition 14. Let γ ∈ B̂ be any graph. The integral

b_{γ}(X, σ) := ∫_{X} RW_{X,σ}(γ) exp(σ + σ̄) (74)

is called the Rozansky-Witten invariant of (X, σ) associated to γ.

5.2. Examples and properties of Rozansky-Witten classes. We summarise in this subsection the properties of the Rozansky-Witten classes that will be of use for us. For proofs take a look at [10], please. Let (X, σ) again be a holomorphic symplectic manifold.

Proposition 15. The map RW_{X,σ} : B̂ → H^{∗,∗}(X) is a morphism of graded algebras.

Proposition 16. For all γ ∈ B̂′ and γ′ ∈ B̂ we have

RW_{X,σ}(⟨γ, γ′⟩) = ⟨RW_{X,σ}(γ), RW_{X,σ}(γ′)⟩ . (75)

Example 9. The cohomology class [σ] ∈ H^{2,0}(X) is a Rozansky-Witten class; more precisely, we have

RW_{X,σ}(ℓ) = 2[σ]. (76)

Example 10. The components of the Chern charakter are Rozansky-Witten invariants:

− RW_{X,σ}(w_{2k}) = RW_{X,σ}(w̃_{2k}) = s_{2k}. (77)

The next two proposition actually aren’t stated in [10], so we shall give ideas of their proofs here.

Proposition 17. Let ν : (X, ν^{∗}σ) → (Y, σ) be a Galois cover of holomorphic symplectic manifolds. For every graph homology class γ ∈ B̂,

RW_{X,ν^{∗}σ}(γ) = ν^{∗} RW_{Y,σ}(γ). (78)

Proof. As ν is a Galois cover, we can identify T_{X} with ν^{∗}T_{Y} and so α̃_{X} with ν^{∗}α̃_{Y} where α̃_{X} and α̃_{Y} are the Atiyah classes of X and Y. By definition of the Rozansky-Witten classes, (78) follows. □

Lemma 2. Let (X, σ) and (Y, τ) be two holomorphic symplectic manifolds. If the tangent bundle of Y is trivial,

RW_{X×Y,p^{∗}σ+q^{∗}τ} (γ) = p^{∗} RW_{X,σ}(γ) (79)

for all graphs γ ∈ B̂′. Here p : X × Y → X and q : X × Y → Y denote the canonical projections.

Proof. This lemma is a special case of the more general proposition in [16] that relates the coproduct in graph homology with the product of holomorphic symplectic manifolds. Since all Rozansky-Witten classes for graphs with at least one trivalent vertex vanish on Y, our lemma follows easily from J. Sawon’s statement. □

5.3. Rozansky-Witten classes of closed graphs. Let γ be a homogeneous closed graph of degree 2k. For every compact holomorphic symplectic manifold (X, σ), we have RW_{X,σ}(γ) ∈ H^{0,2k}(X). If X is irreducible, we therefore have RW_{X,σ}(γ) = β_{γ} · [σ̄]^{k} for a certain β_{γ} ∈ ℂ. We can express β_{γ} as

(80) β_{γ} = \frac{∫_{X} RW_{X,σ}(γ)σ̄^{n−k}σ^{n}}{∫_{X}(σσ̄)^{n}} = \frac{(n − k)!}{n!} \frac{∫_{X} RW_{X,σ}(γ) exp(σ + σ̄)}{∫_{X} exp(σ + σ̄)}

where 2n is the dimension of X. This formula makes also sense for non-irreducible X, which leads us to the following definition:

Definition 15. Let (X, σ) be a compact holomorphic symplectic manifold (X, σ) of dimension 2n. For any homogeneous closed graph homology class γ of degree 2k with k ≤ n we set

(81) β_{γ}(X, σ) := \frac{(n − k)!}{n!} \frac{∫_{X} RW_{X,σ}(γ) exp(σ + σ̄)}{∫_{X} exp(σ + σ̄)}

By linear extension, we can define β_{γ}(X, σ) also for non-homogeneous closed graph homology classes γ.

Remark 12. The map ^{t}B̂ → ℂ, γ ↦ β_{γ}(X, σ) is linear. If X is irreducible, it is also a homomorphism of rings.

For polywheels w̃_{2λ}, we can express β_{\left⟨{ \atop w̃_{2λ}}\right⟩} in terms of characteristic classes:

Proposition 18. Let (X, σ) be a compact holomorphic symplectic manifold of dimension 2n and k ∈ {1, …, n}. Let λ ∈ P (k) be any partition of k. Then

(82) ∫_{X} RW_{X,σ}(\left⟨{ \atop w̃_{2λ}}\right⟩) exp(σ + σ̄) = ∫_{X} s_{2λ}(X) exp(σ + σ̄).

Proof. We calculate

(83) ∫_{X} RW_{X,σ}(\left⟨{ \atop w̃_{2λ}}\right⟩) exp(σ + σ̄) = ∫_{X} RW_{X,σ}(⟨w̃_{2λ}, exp(ℓ/2)⟩) exp(σ + σ̄)

= ∫_{X} ⟨s_{2λ}, exp σ⟩ exp(σ + σ̄) = ∫_{X} s_{2λ} exp(σ + σ̄).

□

6. Calculation for the example series

6.1. Proof of the main theorem. Let X be a smooth projective surface that admits a holomorphic symplectic form (e.g. a K3 surface or an abelian surface). Let us fix a holomorphic symplectic form σ ∈ H^{2,0}(X) that is normalised such that ∫_{X} σσ̄ = 1. It is known ([9]) that X^{[n]} for all n ∈ ℕ_{0} is a compact holomorphic symplectic manifold. For every homogeneous closed graph homology class γ of degree 2k and every n ∈ ℕ_{0}, we set

h_{γ}^{X}(n) := β_{γ}(X^{[k+n]}, σ^{[k+n]}). (84)

By linear extension, we define h_{γ}^{X}(n) for non-homogeneous closed graph homology classes γ.

Proposition 19. For all closed graph homology classes γ, we have

∑_{n=0}^{∞} \frac{q^{n}}{n!} h_{γ}^{X}(n) = ∑_{l=0}^{∞} ∫_{X^{[l]}} RW_{X^{[l]},σ^{[l]}}(γ) exp(q^{\frac{1}{2}} (σ^{[l]} + σ̄^{[l]})) (85)

in ℂ[[q]].

Proof. Let us assume that γ is homogeneous of degree 2k. Then

h_{γ}^{X}(n) = β_{γ}(X^{[k+n]}, σ^{[k+n]}) =

\frac{n!}{(n + k)!} \frac{∫_{X^{[k+n]}} RW_{X^{[k+n]},σ^{[k+n]}}(γ) exp(σ^{[k+n]} + σ̄^{[k+n]})}{∫_{X^{[k+n]}} exp(σ^{[k+n]} + σ̄^{[k+n]})}

= n! ∫_{X^{[k+n]}} RW_{X^{[k+n]},σ^{[k+n]}}(γ) exp(σ + σ̄).

In the last equation we have used Corollary 1. Summing up and introducing the counting parameter q yields the claim. □

Proposition 20. Let a_{2}, a_{4}, … be formal parameters. We set

ω(t) := ∑_{k=1}^{∞} a_{2k}t^{k}w̃_{2k} ∈ B̂^{1}[a_{2}, a_{4}, …][t] (86)

and call ω the universal wheel. Further, we set W (t) := exp(ω(t)) and W := W (1). The Rozansky-Witten classes of the universal wheel are encoded by

∑_{n=0}^{∞} \frac{q^{n}}{n!} h_{⟨W (t)⟩}^{X}(n) = exp(qA(t)) exp(c_{2}(X)D(t)). (87)

Proof. Using Proposition 19 and Proposition 18 yields:

∑_{n=0}^{∞} \frac{q^{n}}{n!} h_{⟨W (t)⟩}^{X}(n) = ∑_{l=0}^{∞} ∫_{X^{[l]}} RW_{X^{[l]},σ^{[l]}}(⟨W (t)⟩) exp(q^{\frac{1}{2}} (σ^{[l]} + σ̄^{[l]}))

= ∑_{l=0}^{∞} ∫_{X^{[l]}} exp (∑_{k=1}^{∞} a_{2k}s_{2k}(X^{[l]})t^{k}) exp(q^{\frac{1}{2}} (σ^{[l]} + σ̄^{[l]}))

= exp(qA(t)) exp(c_{2}(X)D(t)).
□

Corollary 2. For every n ∈ ℕ_{0} we have

(88) h_{⟨W (t)⟩}^{X}(n) = exp(c_{2}(X)D(t)) exp(n log A(t))

Proof. Comparision of coefficients in (87) gives

h_{⟨W (t)⟩}^{X}(n) = A(t)^{n} exp(c_{2}(X)D(t)).

Lastly, note that A is a power series in t that has constant coefficient one. □

Remark 13. By equation (88) we shall extend the definition of h_{⟨W (t)⟩}^{X}(n) to all n ∈ ℤ.

Proposition 21. Let A be an abelian surface. Let us fix a holomorphic symplectic form σ ∈ H^{2,0}(A) that is normalised such that ∫_{A} σσ̄ = 1. Let γ be a homogeneous connected closed graph of degree 2k. Then we have

β_{γ}(A^{[[n]]}, σ^{[[n]]}) = \frac{n}{n − k} β_{γ}(A^{[n]}, σ^{[n]}) (89)

for any n > k.

Proof. The proof is a straight-forward calculation:

β_{γ}(A^{[[n]]}, σ^{[[n]]}) = \frac{(n − 1 − k)!}{(n − 1)!} \frac{∫_{A^{^{[}^{[n]}^{]}}} RW_{A^{^{[}^{[n]}^{]}},σ^{^{[}^{[n]}^{]}}}(γ) exp(σ^{[[n]]} + σ̄^{[[n]]})}{∫_{A^{^{[}^{[n]}^{]}}} exp(σ^{[[n]]} + σ̄^{[[n]]})}

= \frac{(n − 1 − k)!}{(n − 1)!} \frac{∫_{A^{^{[}^{[n]}^{]}}} RW_{A^{^{[}^{[n]}^{]}},σ^{^{[}^{[n]}^{]}}}(γ) exp(σ^{[[n]]} + σ̄^{[[n]]})}{∫_{A^{^{[}^{[n]}^{]}}} exp(σ^{[[n]]} + σ̄^{[[n]]}} \frac{∫_{A} exp(nσ + nσ̄)}{∫_{A} exp(nσ + nσ̄)}

= \frac{(n − 1 − k)!}{(n − 1)!} \frac{∫_{A^{[n]}} RW_{A^{[n]},σ^{[n]}}(γ) exp(σ^{[n]} + σ̄^{[n]})}{∫_{A^{[n]}} exp(σ^{[n]} + σ̄^{[n]})} = \frac{n}{n − k} β_{γ}(A^{[n]}, σ^{[n]}),

where we have used Proposition 12, Proposition 17 and Lemma 2. □

Documenta Mathematica 8 (2003) 591–623

Theorem 3. For any homogenous connected closed graph of degree 2k lying in the algebra C of polywheels there exist two rational numbers a_{γ}, c_{γ} such that for each K3 surface X together with a symplectic form σ ∈ H^{2,0}(X) with ∫_{X} σσ̄ = 1 and n ≥ k we have

β_{γ}(X^{[n]}, σ^{[n]}) = a_{γ}n + c_{γ} (90)

and that for each abelian surface A together with a symplectic form σ ∈ H^{2,0}(X) with ∫_{X} σσ̄ = 1 and n > k we have

β_{γ}(A^{[[n]]}, σ^{[[n]]}) = a_{γ}n. (91)

Proof. Let (X, σ) be a K3 surface or an abelian surface together with a symplectic form with ∫_{X} σσ̄ = 1. Let W_{2k} be the homogeneous component of degree 2k of W (1). Then W (t) = ∑_{k=0}^{∞} W_{2k}t^{k}. Thus we have by (88):

h_{⟨W (t)⟩}^{X}(n) = ∑_{k=0}^{∞} h_{⟨W_{2k}⟩}^{X}(n)t^{k} = U_{c_{2}(X)}(t) exp(nV (t)) (92)

with U_{c_{2}(X)}(t) := exp(c_{2}(X)D(t)) and V (t) := log A(t). Let us consider the case of a K3 surface X first. Note that c_{2}(X) = 24. By definition of h_{γ}^{X}(n) we have

β_{⟨W_{2k}⟩}(X^{[n]}, σ^{[n]}) = h_{⟨W_{2k}⟩}^{X}(n − k) (93)

for all n ≥ k. For n < k we take this equation as a definition for its left hand side. Let the power series T (t) ∈ ℚ[a_{2}, a_{4}, …][[t]] be defined by T (t exp(V (t))) = t, and set Ṽ (t) := V (T (t)) and Ũ := \frac{U_{24}(T (t))}{1+T (t)V′(T (t))}. By Lemma 1, we have

β_{⟨W (t)⟩}(X^{[n]}, σ^{[n]}) = ∑_{k=0}^{∞} h_{⟨W_{2k}⟩}^{X}(n − k)t^{k} = Ũ(t) exp(nṼ (t)).

Note that W (t) is of the form exp(γ) where γ is a connected graph. By Proposition 6 and Remark 12 we therefore have

β_{⟨⟨W (t)⟩⟩}(X^{[n]}, σ^{[n]}) = β_{log ⟨W (t)⟩}(X^{[n]}, σ^{[n]}) = log β_{⟨W (t)⟩} = nṼ (t) + log U(t).

Finally, let λ be any partition. Setting

∂_{2λ} := (∏_{i=1}^{∞} \frac{∂^{λ_{i}}}{∂a_{i}^{λ_{i}}} |_{a_{i}=0})|_{t=0} .

It is

β_{⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩} = ∂_{2λ}β_{⟨⟨W (t)⟩⟩} = n∂_{2λ}Ṽ (t) + ∂_{2λ} log Ũ(t),

so the theorem is proven for K3 surfaces and all connected graph homology classes of the form ⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩ and thus for all connected graph homology classes in C.

Let us now turn to the case of a generalised Kummer variety, i.e. let X = A be an abelian surface and n ≥ 1. Note that c_{2}(A) = 0. Here, we have due to Proposition 21:

β_{⟨W_{2k}⟩}(A^{[[n]]}, σ^{[[n]]}) = \frac{n}{n − k} h_{⟨W_{2k}⟩}^{A}(n − k)

for n > k. For n ≤ k we take this equation as a definition for its left hand side. As U_{0}(t) = 1, Lemma 1 yields in this case that

β_{⟨W (t)⟩}(A^{[[n]]}, σ^{[[n]]}) = ∑_{k=0}^{∞} \frac{n}{n − k} h_{⟨W_{2k}⟩}^{X}(n − k)t^{k} = exp nṼ (t).

We can then proceed as in the case of the Hilbert scheme of a K3 surface to finally get

β_{⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩} = n∂_{2λ}(Ṽ (t)).
□

6.2. Some explicit calculations. Now, we’d like to calculate the constants a_{γ} and c_{γ} for any homogeneous connected closed graph homology class γ of degree 2k lying in C. By the previous theorem, we can do this by calculating β_{γ} on (X, σ) for (X, σ) being the 2k-dimensional Hilbert scheme of points on a K3 surface and the 2k-dimensional generalised Kummer variety. We can do this by recursion over k: Let the calculation having been done for homogeneous connected closed graph homology classes γ of degree less than 2k in C and both example series. Let λ be any partition of k. We can express ⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩ as

⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩ = \left⟨{ \atop w̃_{2λ}}\right⟩ + P, (94)

where P is a polynomial in homogeneous connected closed graph homology classes γ of degree less than 2k in C (for this see Proposition 6). Therefore, β_{⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩}(X, σ) is given by

β_{⟨\left⟨{ \atop w̃_{2λ}}\right⟩⟩}(X, σ) = β_{\left⟨{ \atop w̃_{2λ}}\right⟩}(X, σ) + P′, (95)

where P′ is a polynomial in terms like β_{γ′}(X, σ) with γ′ ∈ C and deg γ′ < 2k. However, these terms have been calculated in previous recursion steps. Therefore, the only thing new we have to calculate in this recursion step is β_{\left⟨{ \atop w̃_{2λ}}\right⟩}(X, σ). We have:

(96) β_{\left⟨{ \atop w̃_{2λ}}\right⟩}(X, σ) = \frac{1}{k!} \frac{∫_{X} RW_{X,σ}(w̃_{2λ}) exp(σ + σ̄)}{∫_{X} exp(σ + σ̄)} = \frac{∫_{X} s_{2λ}(X)}{∫_{X} exp (σ + σ̄)}.

As all the Chern numbers of X can be computed with the help of Bott’s residue formula (see [3] for the case of the Hilbert scheme and [11] for the case of the generalised Kummer variety), we therefore are able to calculate β_{\left⟨{ \atop w̃_{2λ}}\right⟩}(X, σ). This ends the recursion step as we have given an algorithm to compute a_{γ} and c_{γ} for any homogeneous connected closed graph homology class γ of degree 2k in C.

We worked through the recursion for k = 1, 2, 3. Firstly, we have

\begin{aligned}⟨\left⟨{ \atop w̃_{2}}\right⟩⟩ & = \left⟨{ \atop w̃_{2}}\right⟩ \\ ⟨⟨w̃_{2}^{2}⟩⟩ & = \left⟨{ \atop w̃_{4}}\right⟩ − ⟨\left⟨{ \atop w̃_{2}}\right⟩⟩^{2} \\ ⟨\left⟨{ \atop w̃_{4}}\right⟩⟩ & = \left⟨{ \atop w̃_{4}}\right⟩ \\ ⟨⟨w̃_{2}^{3}⟩⟩ & = ⟨w̃_{2}^{3}⟩ − 3 ⟨\left⟨{ \atop w̃_{2}}\right⟩⟩ ⟨⟨w̃_{2}^{2}⟩⟩ − ⟨\left⟨{ \atop w̃_{2}}\right⟩⟩^{3} \\ ⟨⟨w̃_{2}w̃_{4}⟩⟩ & = ⟨w̃_{2}w̃_{4}⟩ − ⟨\left⟨{ \atop w̃_{2}}\right⟩⟩ ⟨\left⟨{ \atop w̃_{4}}\right⟩⟩ \\ ⟨\left⟨{ \atop w̃_{6}}\right⟩⟩ & = \left⟨{ \atop w̃_{6}}\right⟩ . \end{aligned} (97)

Not let X be a K3 surface and A an abelian surface. Let us denote by σ either a holomorphic symplectic two-form on X with ∫_{X} σσ̄ = 1 or on A with ∫_{A} σσ̄ = 1. We use the following table of Chern numbers for the Hilbert scheme of points on a K3 surface:

k s  s[X^{[k]}] s[A^{[[k+1]]}]
1 s_{2}  -48  -48
2 s_{2}^{2}  3312  3024
s_{4}  360  1080
3 s_{2}^{3} -294400 -241664
s_{2}s_{4} -29440  -66560
s_{6} -4480 -22400

Going through the recursion, we arrive at the following table:

k  γ  β_{γ}(A^{[[k+1]]}) β_{γ}(X^{[k]})  a_{γ} c_{γ}
1 ⟨\left⟨{ \atop w̃_{2}}\right⟩⟩  -24  -48  12 -36
2 ⟨⟨w̃_{2}^{2}⟩⟩  -288  -288  -96 -96
⟨\left⟨{ \atop w̃_{4}}\right⟩⟩  360  360 120 120
3 ⟨⟨w̃_{2}^{3}⟩⟩  -5120  -4096 -1280 -256
⟨⟨w̃_{2}w̃_{4}⟩⟩  6400  5120 1600 320
⟨\left⟨{ \atop w̃_{6}}\right⟩⟩  -5600  -4480 -1400 -280

Now, we would like to turn to Rozansky-Witten invariants: Let γ be any homogeneous closed graph homology class of degree 2k. For any holomorphic symplectic manifold (X, σ) of dimension 2n, the associated Rozansky-Witten invariant is given by

(98) b_{γ}(X, σ) = ∫_{X} RW_{X,σ}(γ) exp(σ + σ̄) = \frac{1}{n!(n − k)!}β_{γ}(X, σ) ∫_{X}(σσ̄)^{n}

= \frac{n!}{(n − k)!}β_{γ}(X, σ) ∫_{X} exp(σ + σ̄).

To know the Rozansky-Witten invariant associated to closed graph homology classes, we therefore have just to calculate the value of β_{γ}. On an irreducible holomorphic symplectic manifold, γ ↦ β_{γ} is multiplicative with respect to the disjoint union of graphs, so it is enough to calculate β_{γ} for connected closed graph homology classes. However, we have just done this for the Hilbert schemes of points on a K3 surface and the generalised Kummer varieties — as long as γ is spanned by the connected polywheels. By the procedure outlined above, Theorem 3 therefore enables us to compute all Rozansky-Witten invariants of the two example series associated to closed graph homology classes lying in C.

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Marc A. Nieper-Wißkirchen Johannes Gutenberg-Universität Mainz Fachbereich für Mathematik und Informatik 55099 Mainz Germany marc@nieper-wisskirchen.de 624

Documenta Mathematica 8 (2003)